# Phonons in magnon superfluid and symmetry breaking field

###### Abstract

Recent experiments [1, 2] which measured the spectrum of the Goldstone collective mode of coherently precessing state in He-B are discussed using the presentation of the coherent spin precession in terms of the Bose-Einstein condensation of magnons. The mass in the spectrum of the Goldstone boson – phonon in the superfluid magnon liquid – is induced by the symmetry breaking field, which is played by the RF magnetic field.

Phonons in magnon superfluid and symmetry breaking field \sodtitlePhonons in magnon superfluid and symmetry breaking field \rauthorG. E. Volovik \sodauthorVolovik \dates23 April 2008*

The phase coherent precession of magnetization has been discovered in He-B in 1984 [3, 4]. Recently two experiments with homogeneously precessing domain (HPD) [1, 2] reported the gap in the spectrum of the collective mode of the coherent precession. The gap is proportional to , where is applied RF field. Here we discuss this phenomenon using the description of the coherent precession in terms of the Bose-Einstein condensation (BEC) of magnons [5, 6].

The hydrodynamic energy functional describing the magnon BEC in He-B has the same structure as in any other superfluid system:

(1) |

where the magnon number density and the phase of magnon Bose condensate are canonically conjugated variables; is equal to the phase of the coherent precession of magnetization; is the superfluid velocity of BEC of magnons; is the chemical potential; and is the symmetry breaking term which depends explicitly on . The energy has some peculiar properties for coherent precession in general and for He-B in particular [7].

(i) In the coherent precession, the chemical potential (which is thermodynamically conjugated to magnon density ) is determined by frequencies: , where is precession frequency; is Larmor frequency; is an external magnetic field; and is gyromagnetic ratio of He atom.

(ii) The magnon number density is related to the tilting angle of precession: , where is the equilibrium spin density , and is magnetic susceptibility of He-B.

(iii) Magnons typically have anisotropic mass. In superfluid He-B, longitudinal and transverse masses depend on (and thus on ) [5]:

(2) |

(3) | |||

(4) |

where the parameters and are spin wave velocities in He-B.

(iv) Anisotropic mass of magnon leads to anisotropic superfluid density and superfluid velocity in the magnon BEC:

(5) |

(Note that magnon superfluid velocity has nothing to do with the superfluid velocity of the background He-B liquid: the latter is , where is the mass of He atom, and is the phase of the order parameter in He-B. Magnon mass is typically much smaller than .)

The mass supercurrent transferred by magnon liquid is isotropic:

(6) |

while the spin supercurrent is not:

(7) |

This equation, which demonstrates that the nondissipative spin current is transferred by the mass-current velocity of magnon superfluid, is typical for the fully spin-polarized superfluids. The same occurs in superfluid A phase of He, where only single spin component is superfluid [8]. Our magnon superfluid is also fully polarized, since all the magnons have the same direction of spin.

(v) The magnon interaction energy is provided by spin-orbit (dipole-dipole) interaction, which has a very peculiar form in He-B:

(8) |

where is Heaviside step function; is Leggett frequency (we assume that ). Stable coherent precession (HPD) occurs only at large enough magnon density , where . This corresponds to .

(vi) Finally, the symmetry-breaking term is induced by the RF field , which is transverse to the applied constant field and serves as a source of magnons. In continuous wave NMR experiments the RF field prescribes the frequency of precession, , and thus fixes the chemical potential ; while in the state of free precession the chemical potential is determined by the number of pumped magnons. depends explicitly on the phase of precession with respect to the direction of the RF-field in the precessing frame:

(9) |

In terms of the macroscopic wave function of magnon Bose condensate it has the form:

(10) |

where the symmetry-breaking field is

(11) |

The hydrodynamic energy in terms of the canonically conjugated hydrodynamic variables and allows us to find the spectrum the low frequency collective modes of the coherent precession. In the absence of the RF field there is a Goldstone mode coming from the degeneracy of the precessing states with respect to the condensate phase . This is the sound wave (phonon) in magnon superfluid. The sound wave velocity is determined by compressibility of magnon superfluid and by magnon mass. Since the mass is anisotropic the phonon spectrum is also anisotropic:

(12) |

In the typical experiments with HPD, is close to 1/4, i.e. . For such one has:

(13) |

(14) |

The sound in magnon subsystem propagating along the field with the speed in Eq.(13) has been calculated in Ref.[9] and observed in Ref. [10].

The Goldstone boson (phonon) acquires mass (gap) due to the transverse RF field . The latter plays the role of the symmetry breaking field, since it violates the symmetry of precession. The symmetry breaking term in Eq. (9) adds the isotropic mass (gap) to the phonon spectrum. For one obtains:

(15) |

This gap has been first calculated in Ref. [1] using general Leggett equations for spin dynamics in He-B. It was measured in Refs. [1, 2].

I thank V.V. Dmitriev and V.B. Eltsov for discussion. This work was supported in part by the Russian Foundation for Basic Research (grant 06-02-16002-a).

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